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41: 2.3 Integrals of a Real Variable

… ►Then … ►Then the series obtained by substituting (2.3.7) into (2.3.1) and integrating formally term by term yields an asymptotic expansion: … ►When the parameter $x$ is large the contributions from the real and imaginary parts of the integrand in …However, cancellation does not take place near the endpoints, owing to lack of symmetry, nor in the neighborhoods of zeros of $p^{\prime }(t)$ because $p(t)$ changes relatively slowly at these stationary points. … ►A uniform approximation can be constructed by quadratic change of integration variable: …

42: 10.17 Asymptotic Expansions for Large Argument

… ►

10.17.1 $$a_k(\nu )=\frac{(4{\nu }^2-1^2)(4{\nu }^2-3^2)\mathrm{\cdots }(4{\nu }^2-{(2k-1)}^2)}{k!8^k}=\frac{{\left(\frac{1}{2}-\nu \right)}_k{\left(\frac{1}{2}+\nu \right)}_k}{{(-2)}^{k}k!},$$ $k\ge 1$,

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Defines:

$a_k(\nu )$: polynomial coefficient (locally)

Symbols:

${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $!$: factorial (as in $n!$), $k$: nonnegative integer and $\nu$: complex parameter

Referenced by:

§10.49(i), Erratum (V1.1.3) for Additions

Permalink:

http://dlmf.nist.gov/10.17.E1

Encodings:

pMML, png, TeX

Addition (effective with 1.1.3):

An alternative Pochhammer symbol representation was added.

See also:

Annotations for §10.17(i), §10.17 and Ch.10

►

10.17.2 $$\omega =z-\frac{1}{2}\nu \pi -\frac{1}{4}\pi ,$$

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Defines:

$\omega$ (locally)

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $z$: complex variable and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.17.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(i), §10.17 and Ch.10

… ►

10.17.8 $$b_k(\nu )=\frac{\left((4{\nu }^2-1^2)(4{\nu }^2-3^2)\mathrm{\cdots }(4{\nu }^2-{(2k-3)}^2)\right)(4{\nu }^2+4k^2-1)}{k!8^k}.$$

ⓘ

Defines:

$b_k(\nu )$: polynomial coefficient (locally)

Symbols:

$!$: factorial (as in $n!$), $k$: nonnegative integer and $\nu$: complex parameter

Permalink:

http://dlmf.nist.gov/10.17.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(ii), §10.17 and Ch.10

… ►where $𝒱$ denotes the variational operator (2.3.6), and the paths of variation are subject to the condition that $|\mathrm{\Im }t|$ changes monotonically. … ►

10.17.17 $$R_{\mathrm{\ell }}^{\pm }(\nu ,z)={(-1)}^{\mathrm{\ell }}2\cos \left(\nu \pi \right)\left(\sum _{k=0}^{m-1}{(\pm \mathrm{i})}^k\frac{a_k(\nu )}{z^k}{G}_{\mathrm{\ell }-k}\left(\mp 2\mathrm{i}z\right)+R_{m,\mathrm{\ell }}^{\pm }(\nu ,z)\right),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $\mathrm{i}$: imaginary unit, $G_p\left(z\right)$: rescaled terminant function, $m$: integer, $k$: nonnegative integer, $z$: complex variable, $\nu$: complex parameter, $a_k(\nu )$: polynomial coefficient and $R_{\mathrm{\ell }}^{\pm }(\nu ,z)$: remainder

Permalink:

http://dlmf.nist.gov/10.17.E17

Encodings:

pMML, png, TeX

See also:

Annotations for §10.17(v), §10.17 and Ch.10

…

43: 19.28 Integrals of Elliptic Integrals

… ► ►

19.28.1 $${\int }_0^1{t}^{\sigma -1}{R}_F(0,t,1)dt=\frac{1}{2}{\left(\mathrm{B}(\sigma ,\frac{1}{2})\right)}^2,$$

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Symbols:

$R_F(x,y,z)$: symmetric elliptic integral of first kind, $\mathrm{B}(a,b)$: beta function, $dx$: differential of $x$ , $\int$: integral and $\sigma$: parameter

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

►

19.28.2 $${\int }_0^1{t}^{\sigma -1}{R}_G(0,t,1)dt=\frac{\sigma }{4\sigma +2}{\left(\mathrm{B}(\sigma ,\frac{1}{2})\right)}^2,$$

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Symbols:

$R_G(x,y,z)$: symmetric elliptic integral of second kind, $\mathrm{B}(a,b)$: beta function, $dx$: differential of $x$ , $\int$: integral and $\sigma$: parameter

Permalink:

http://dlmf.nist.gov/19.28.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

►

19.28.3 $${\int }_0^1{t}^{\sigma -1}(1-t)R_D(0,t,1)dt=\frac{3}{4\sigma +2}{\left(\mathrm{B}(\sigma ,\frac{1}{2})\right)}^2.$$

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Symbols:

$R_D(x,y,z)$: elliptic integral symmetric in only two variables, $\mathrm{B}(a,b)$: beta function, $dx$: differential of $x$ , $\int$: integral and $\sigma$: parameter

Referenced by:

§19.28, §19.28

Permalink:

http://dlmf.nist.gov/19.28.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

►

19.28.4 $${\int }_0^1{t}^{\sigma -1}{(1-t)}^{c-1}{R}_{-a}(b_1,b_2;t,1)dt=\frac{\mathrm{\Gamma }\left(c\right)\mathrm{\Gamma }\left(\sigma \right)\mathrm{\Gamma }\left(\sigma +b_2-a\right)}{\mathrm{\Gamma }\left(\sigma +c-a\right)\mathrm{\Gamma }\left(\sigma +b_2\right)},$$ $c=b_1+b_2>0$, $\mathrm{\Re }\sigma >\max (0,a-b_2)$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\int$: integral, $\mathrm{\Re }$: real part and $\sigma$: parameter

Referenced by:

§19.28

Permalink:

http://dlmf.nist.gov/19.28.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §19.28 and Ch.19

…

44: 28.6 Expansions for Small $q$

… ►

28.6.1 $$a_0\left(q\right)=-\frac{1}{2}{q}^2+\frac{7}{128}{q}^4-\frac{29}{2304}{q}^6+\frac{68687}{\mathrm{188\hspace{0.33em}74368}}{q}^8+\mathrm{\cdots },$$

ⓘ

Symbols:

$a_n\left(q\right)$: eigenvalues of Mathieu equation and $q=h^2$: parameter

A&S Ref:

20.2.25

Referenced by:

§28.6(i)

Permalink:

http://dlmf.nist.gov/28.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(i), §28.6 and Ch.28

… ►

28.6.4 $$a_2\left(q\right)=4+\frac{5}{12}{q}^2-\frac{763}{13824}{q}^4+\frac{\mathrm{10\hspace{0.33em}02401}}{\mathrm{796\hspace{0.33em}26240}}{q}^6-\frac{\mathrm{16690\hspace{0.33em}68401}}{\mathrm{45\hspace{0.33em}86471\hspace{0.33em}42400}}{q}^8+\mathrm{\cdots },$$

ⓘ

Symbols:

$a_n\left(q\right)$: eigenvalues of Mathieu equation and $q=h^2$: parameter

Permalink:

http://dlmf.nist.gov/28.6.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(i), §28.6 and Ch.28

►

28.6.5 $$b_2\left(q\right)=4-\frac{1}{12}{q}^2+\frac{5}{13824}{q}^4-\frac{289}{\mathrm{796\hspace{0.33em}26240}}{q}^6+\frac{21391}{\mathrm{45\hspace{0.33em}86471\hspace{0.33em}42400}}{q}^8+\mathrm{\cdots },$$

ⓘ

Symbols:

$b_n\left(q\right)$: eigenvalues of Mathieu equation and $q=h^2$: parameter

Permalink:

http://dlmf.nist.gov/28.6.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(i), §28.6 and Ch.28

… ►

28.6.8 $$a_4\left(q\right)=16+\frac{1}{30}{q}^2+\frac{433}{\mathrm{8\hspace{0.33em}64000}}{q}^4-\frac{5701}{\mathrm{27216\hspace{0.33em}00000}}{q}^6+\mathrm{\cdots },$$

ⓘ

Symbols:

$a_n\left(q\right)$: eigenvalues of Mathieu equation and $q=h^2$: parameter

Permalink:

http://dlmf.nist.gov/28.6.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §28.6(i), §28.6 and Ch.28

… ►For the corresponding expansions of ${\mathrm{se}}_m(z,q)$ for $m=3,4,5,\mathrm{\dots }$ change $\cos$ to $\sin$ everywhere in (28.6.26). …

45: 3.5 Quadrature

… ►

3.5.39 $$g(t)=\frac{1}{2\pi \mathrm{i}}{\int }_{\sigma -\mathrm{i}\mathrm{\infty }}^{\sigma +\mathrm{i}\mathrm{\infty }}{\mathrm{e}}^{tp}G(p)dp,$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $g(t)$: function, $G(p)$: Laplace transform of $g(t)$ and $\sigma$: parameter

A&S Ref:

29.2.1 (in different form)

Referenced by:

§3.5(viii)

Permalink:

http://dlmf.nist.gov/3.5.E39

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(viii), §3.5(viii), §3.5 and Ch.3

… ►The integral (3.5.39) has the form (3.5.35) if we set $\zeta =tp$, $c=t\sigma$, and $f(\zeta )=t^{-1}{\zeta }^{s}G(\zeta /t)$. … ►When $\lambda$ is large the integral becomes exponentially small, and application of the quadrature rule of §3.5(viii) is useless. … ►

3.5.43 $$\mathrm{erfc}\lambda \sim \frac{{\mathrm{e}}^{-{\lambda }^2}}{\sqrt{\pi }\lambda },$$ $\lambda \to \mathrm{\infty }$.

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Symbols:

$\sim$: asymptotic equality, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{erfc}z$: complementary error function, $\mathrm{e}$: base of natural logarithm and $\lambda$: parameter

Permalink:

http://dlmf.nist.gov/3.5.E43

Encodings:

pMML, png, TeX

See also:

Annotations for §3.5(ix), §3.5(ix), §3.5 and Ch.3

… ►

3.5.46 $$\mathscr{H}f\left(x\right)=\frac{1}{\pi }{⨍}_{-\mathrm{\infty }}^{\mathrm{\infty }}\frac{f(t)}{t-x}dt,$$ $x\in ℝ$,

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Symbols:

$\mathscr{H}\left(f\right)\left(x\right)$: Hilbert transform, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\in$: element of, ${⨍}_a^b$: Cauchy principal value and $ℝ$: real line

Permalink:

http://dlmf.nist.gov/3.5.E46

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

The notation for the Hilbert transform was changed to $\mathscr{H}f\left(x\right)$ from $\mathscr{H}(f;x)$.

See also:

Annotations for §3.5(ix), §3.5(ix), §3.5 and Ch.3

…

46: Bibliography K

… ►

Connecting Jacobi elliptic functions with different modulus parameters. Pramana 63 (5), pp. 921–936.

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External Links:

Pdf, Document

Referenced by:

§22.7(iii).

Encodings:

BibTeX

… ►

A vortex filament moving without change of form. J. Fluid Mech. 112, pp. 397–409.

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External Links:

ISSN 0022-1120, Document, MathReview, ZentralBlatt

Referenced by:

§19.35(ii).

Encodings:

BibTeX

… ►

Approximation Formulae for Generalized Hypergeometric Functions for Large Values of the Parameters. J. B. Wolters, Groningen.

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External Links:

MathReview (L. J. Slater), ZentralBlatt

Referenced by:

§16.11(iii).

Encodings:

BibTeX

…

47: 27.2 Functions

… ►

27.2.3 $$\pi \left(x\right)\sim \frac{x}{\ln x}.$$

ⓘ

Symbols:

$\sim$: asymptotic equality, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number and $x$: real number

Referenced by:

§25.16(i), §27.11

Permalink:

http://dlmf.nist.gov/27.2.E3

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►

27.2.4 $$p_n\sim n\ln n.$$

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Symbols:

$\sim$: asymptotic equality, $\ln z$: principal branch of logarithm function, $n$: positive integer and $p,p_1,\mathrm{\dots }$: prime numbers

Permalink:

http://dlmf.nist.gov/27.2.E4

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►

27.2.10 $${\sigma }_{\alpha }\left(n\right)=\sum _{d|n}{d}^{\alpha },$$

ⓘ

Defines:

${\sigma }_{\alpha }\left(n\right)$: sum of powers of divisors of a number

Symbols:

$d$: positive integer, $n$: positive integer and $\alpha$: parameter

A&S Ref:

24.3.3 I.A

Referenced by:

§27.14(ii)

Permalink:

http://dlmf.nist.gov/27.2.E10

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

►is the sum of the $\alpha$th powers of the divisors of $n$, where the exponent $\alpha$ can be real or complex. … ►

27.2.14 $$\mathrm{\Lambda }\left(n\right)=\ln p,$$ $n=p^a$,

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Defines:

$\mathrm{\Lambda }\left(n\right)$: Mangoldt’s function

Symbols:

$\ln z$: principal branch of logarithm function, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $a$: primitive root

Permalink:

http://dlmf.nist.gov/27.2.E14

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.2(i), §27.2 and Ch.27

…

48: 28.31 Equations of Whittaker–Hill and Ince

… ►When $p$ is a nonnegative integer, the parameter $\eta$ can be chosen so that solutions of (28.31.3) are trigonometric polynomials, called Ince polynomials. … ►

28.31.11

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Defines:

$S_p^m(z,NVarxi)$: Ince polynomials (locally)

Symbols:

$m$: integer, $n$: integer, $x$: real variable, $z$: complex variable, $a$: parameter and $\xi$: variable

Referenced by:

§28.31(iii)

Permalink:

http://dlmf.nist.gov/28.31.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §28.31(ii), §28.31 and Ch.28

… ►

28.31.18 $$w^{\prime \prime }+\left(\eta -\frac{1}{8}{\xi }^2-(p+1)\xi \cos \left(2z\right)+\frac{1}{8}{\xi }^2\cos \left(4z\right)\right)w=0,$$

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Symbols:

$\cos z$: cosine function, $z$: complex variable, $w(z)$: Mathieu’s equation solution and $\eta$: change of variable

Permalink:

http://dlmf.nist.gov/28.31.E18

Encodings:

pMML, png, TeX

See also:

Annotations for §28.31(iii), §28.31 and Ch.28

… ►For change of sign of $\xi$, …

49: 10.21 Zeros

… ►where $t$ is a parameter, then ►

10.21.5 $${𝒞}_{\nu }^{\prime }\left({\rho }_{\nu }\right)={𝒞}_{\nu -1}\left({\rho }_{\nu }\right)=-{𝒞}_{\nu +1}\left({\rho }_{\nu }\right).$$

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Symbols:

${𝒞}_{\nu }\left(z\right)$: cylinder function, $\nu$: complex parameter and ${\rho }_{\nu }(t)$: zero of cylinder function

A&S Ref:

9.5.4

Referenced by:

§10.21(ii)

Permalink:

http://dlmf.nist.gov/10.21.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(ii), §10.21 and Ch.10

… ►The parameter $t$ may be regarded as a continuous variable and ${\rho }_{\nu }$, ${\sigma }_{\nu }$ as functions ${\rho }_{\nu }(t)$, ${\sigma }_{\nu }(t)$ of $t$. … ►

10.21.21 $$j_{\nu ,m}^{\prime \prime }\sim c-\frac{\mu +7}{8c}-\frac{28{\mu }^2+424\mu +1724}{3{(8c)}^3}-\mathrm{\cdots },$$

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Symbols:

$\sim$: Poincaré asymptotic expansion, $m$: integer, $\nu$: complex parameter and $c$: zero of cylinder function

Permalink:

http://dlmf.nist.gov/10.21.E21

Encodings:

pMML, png, TeX

See also:

Annotations for §10.21(vi), §10.21 and Ch.10

… ►Corresponding uniform approximations for $y_{\nu ,m}$, $Y_{\nu }^{\prime }\left(y_{\nu ,m}\right)$, $y_{\nu ,m}^{\prime }$, and $Y_{\nu }\left(y_{\nu ,m}^{\prime }\right)$, are obtained from (10.21.41)–(10.21.44) by changing the symbols $j$, $J$, $\mathrm{Ai}$, ${\mathrm{Ai}}^{\prime }$, $a_m$, and $a_m^{\prime }$ to $y$, $Y$, $-\mathrm{Bi}$, $-{\mathrm{Bi}}^{\prime }$, $b_m$, and $b_m^{\prime }$, respectively. …

50: Bibliography B

… ►

A general program to calculate atomic continuum processes using the R-matrix method. Comput. Phys. Comm. 8 (3), pp. 149–198.

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External Links:

Document, Code

Referenced by:

2nd item.

Encodings:

BibTeX

… ►

Approximating the matrix Fisher and Bingham distributions: Applications to spherical regression and Procrustes analysis. J. Multivariate Anal. 41 (2), pp. 314–337.

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External Links:

ISSN 0047-259X, Document, MathReview (A. L. Rukhin), ZentralBlatt

Referenced by:

§35.10, §35.9.

Encodings:

BibTeX

… ►

Mathieu’s Equation for Complex Parameters. Tables of Characteristic Values. U.S. Government Printing Office, Washington, D.C..

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External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

… ►

Table of characteristic values of Mathieu’s equation for large values of the parameter. J. Washington Acad. Sci. 45 (6), pp. 166–196.

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External Links:

MathReview (L. Fox)

Referenced by:

3rd item.

Encodings:

BibTeX

…